Question: Solve for $k$, $ \dfrac{8}{5k - 10} = -\dfrac{4}{k - 2} + \dfrac{k + 1}{4k - 8} $
Answer: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $5k - 10$ $k - 2$ and $4k - 8$ The common denominator is $20k - 40$ To get $20k - 40$ in the denominator of the first term, multiply it by $\frac{4}{4}$ $ \dfrac{8}{5k - 10} \times \dfrac{4}{4} = \dfrac{32}{20k - 40} $ To get $20k - 40$ in the denominator of the second term, multiply it by $\frac{20}{20}$ $ -\dfrac{4}{k - 2} \times \dfrac{20}{20} = -\dfrac{80}{20k - 40} $ To get $20k - 40$ in the denominator of the third term, multiply it by $\frac{5}{5}$ $ \dfrac{k + 1}{4k - 8} \times \dfrac{5}{5} = \dfrac{5k + 5}{20k - 40} $ This give us: $ \dfrac{32}{20k - 40} = -\dfrac{80}{20k - 40} + \dfrac{5k + 5}{20k - 40} $ If we multiply both sides of the equation by $20k - 40$ , we get: $ 32 = -80 + 5k + 5$ $ 32 = 5k - 75$ $ 107 = 5k $ $ k = \dfrac{107}{5}$